Mathematics High School

Verify the following reduction formula:
[tex]\displaystyle \int \sec^n(u)\, du=\frac{\sec^{n-2}(u)\tan(u)}{n-1}+\frac{n-2}{n-1}\int \sec^{n-2}(u)\, du, \; n\neq 1[/tex]

Answers

Answer 1

Answer:

See Explanation.

General Formulas and Concepts:

Pre-Algebra

Distributive PropertyEquality Properties

Algebra I

Combining Like Terms

Algebra II

Exponential Rules

Pre-Calculus

Pythagorean Identities: tan²(x) = sec²(x) - 1

Calculus

Basic Power Rule:

f(x) = cxⁿf’(x) = c·nxⁿ⁻¹

Integration Rule 1:         [tex]\int {cf(x)} \, dx = c\int {f(x)} \, dx[/tex]

Integration Rule 2:        [tex]\int {f(x) \pm g(x)} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]

Integration 1:                 [tex]\int {sec^2(u)} \, du = tan(u) + C[/tex]

Integration by Parts:    [tex]\int {u} \, dv = uv - \int {v} \, du[/tex]

[IBP] LIPET: Logs, inverses, Polynomials, Exponentials, Trig

Step-by-step explanation:

Step 1: Define

[tex]\int {sec^n(u)} \, du[/tex]

Step 2: Rewrite

[Integral - Alg] Separate Exponents:                    [tex]\int {sec^n(u)} \, du = \int {sec^{n-2}(u)sec^2(u)} \, du[/tex]

Step 3: Identify Variables

Using LIPET, we define variables to use IBP.

Use Integration 1.

[tex]u = [sec(u)]^{n-2} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ dv = sec^2(u)du\\du = (n-2)[sec(u)]^{n-3} sec(u)tan(u) \ \ \ \ \ \ \ \ v = tan(u)[/tex]

Step 4: Integrate

Integrate [IBP]:   [tex]\int {sec^n(u)} \, du = tan(u)[sec(u)]^{n-2} - \int [{tan(u)(n-2)[sec(u)]^{n-3}sec(u)tan(u)} ]\, du[/tex][Integral - Alg] Multiply:                  [tex]\int {sec^n(u)} \, du = tan(u)[sec(u)]^{n-2} - \int [{tan^2(u)(n-2)[sec(u)]^{n-2}}] \, du[/tex][integral - Int Rule 1] Simplify:     [tex]\int {sec^n(u)} \, du = tan(u)[sec(u)]^{n-2} - (n-2)\int [{tan^2(u)[sec(u)]^{n-2}}] \, du[/tex][Integral - Pythagorean Identities] Rewrite:   [tex]\int {sec^n(u)} \, du = tan(u)[sec(u)]^{n-2} - (n-2)\int [{[sec^2(u) - 1][sec(u)]^{n-2}}] \, du[/tex][Integral - Alg] Multiply/Distribute:   [tex]\int {sec^n(u)} \, du = tan(u)[sec(u)]^{n-2} - (n-2)\int [{sec^n(u)-[sec(u)]^{n-2}}] \, du[/tex][Integral - Int Rule 2] Rewrite:     [tex]\int {sec^n(u)} \, du = tan(u)[sec(u)]^{n-2} - (n-2) [\int {sec^n(u)} \, du - \int {[sec(u)]^{n-2}} \, du ][/tex][Integral - Alg] Distribute:   [tex]\int {sec^n(u)} \, du = tan(u)[sec(u)]^{n-2} - (n-2) \int {sec^n(u)} \, du + (n-2)\int {[sec(u)]^{n-2}} \, du[/tex]Rewrite:     [tex]\int {sec^n(u)} \, du = sec^{n-2}(u)tan(u) - (n-2) \int {sec^n(u)} \, du + (n-2)\int {[sec(u)]^{n-2}} \, du[/tex][Integral - Alg] Isolate Integral Term:     [tex]\int {sec^n(u)} \, du + (n-2) \int {sec^n(u)} \, du = sec^{n-2}(u)tan(u) + (n-2)\int {[sec(u)]^{n-2}} \, du[/tex][Integral - Alg] Combine Like Terms:     [tex](n - 1)\int {sec^n(u)} \, du = sec^{n-2}(u)tan(u) + (n-2)\int {[sec(u)]^{n-2}} \, du[/tex][Integral 2 - Alg] Rewrite:     [tex](n - 1)\int {sec^n(u)} \, du = sec^{n-2}(u)tan(u) + (n-2)\int {sec^{n-2}(u)} \, du[/tex][Integral - Alg] Isolate Original Integral:     [tex]\int {sec^n(u)} \, du = \frac{sec^{n-2}(u)tan(u)}{n-1} + \frac{n-2}{n-1} \int {sec^{n-2}(u)} \, du[/tex]

And we have proved the Reduction Formula!

Related Questions

An article is sold in Rs 678 at a profit of 13%. Find its cost price.
Please help me

Answers

Answer:

Rs 600.

Step-by-step explanation:

113% is equivalent to Rs 678   So by proportion:

100%  ..   ...   ..  ..    to 678 * 100 / 113

= Rs 600.

Algebra 1, question in the picture below transformations

Answers

Ur correct so yea peepeepoopoo

Two number cubes are rolled and the product of the two numbers shown is calculated. Calculate P(multiple of 6). Record your answer as a fraction, written in lowest terms.

Answers

Answer:

1/6

Step-by-step explanation:

Select all equations that are equivalent to
3(x + 2) = 15

a. 3x + 6 = 15
b. X + 2 = 15
C. X + 2 = 5
d. 3x = 21
e. 3x = 9

Anyone knows the answer please

Answers

the answers are A and E

PLEASE HELP DUE TODAY‼️‼️
————————————
10. Parking Garage A charges $3 per hour for parking. Parking Garage B charges a fee of $2 plus $2 per hour for parking.

Answers

Answer:

garage a y = 3x

garage b y = 2x + 2

just input those equations into desmos it is a website

c (2,6)

in this context the solution means that it costs 6 dollars to say in garage a or b for 2 hours

plz give brainlist

You download 4 songs for $5. Write an equation that uses the constant of of
proportionality to describe the relationship between the cost in d dollars
for s songs.

Answers

Answer:

The equation is;

d = 1.25s

Step-by-step explanation:

Here, we start by getting the price of a single song

From the question;

4 songs = $5

Thus, 1 song will be $5/4 = $1.25

So the proportionality constant is $1.25 per song

To write the relationship, we have;

d = 1.25s

Where d is the cost in dollars and s is the number of songs

If (7^2)^p= 7^6, what is the value of p? 2, 3, 4, 8​

Answers

The value of P Is 3!! Your welcome:))

Suppose a tourist visits Molokai and meets 12 Native Hawaiians, some of whom were strumming ukuleles. If the ratio of those playing ukuleles to those not playing was 1:3, how many Hawaiians did the tourist see playing ukuleles?

Answers

Answer:

4 people

Step-by-step explanation:

4. Write an explicit formula for the sequence 6,2, -2, -6, -10.... Then find the 12th term

Answers

Answer:

-38

Step-by-step explanation:

general formula

a + (n-1)(d)

formula for this question

6 + (n-1)(-4)

find d

2-6=-2-2

-4=-4

d=-4

a=6 d=-4

12th term

T12= 6+(12-1)(-4)

= 6 + (-44)

=-38

hope this helps :)

Solve the inequality. X-4< -5​

Answers

[tex]x-4<-5\\x<-5+4\\x<-1[/tex]

Therefore, x < -1.

Question 9 (5 points)
(02.06 LC)
If one cup is equal to 8 fluid Ounces, how many fluid ounces are in 3 cups?

Answers

Answer:

24 fluid oz

Step-by-step explanation:

1 cup = 8

2 cups = double, or 16

3 cups = triple, or 24

Solve:
4(x-8) > 3 - (2x + 7)

A. 4 2/3

B. 6 2/3

C. 6

D. 7

Answers

Answer:

A. 4 2/3

Step-by-step explanation:

4(x-8) > 3 - (2x + 7)

4x-32 > 3 -2x -7

4x - 32 > -4 -2x

6x -32 > -4

6x > 28

divide both sides by 6

6 divide by 6 is 1

28 divide by 6 is simplified to 14/3 which can be changed into a mixed number 4 2/3

Evaluate:
2x + 5 (y – 1) when x = 8 and y = 5 need answer quick for exams !

Answers

Answer:42

Step-by-step explanation:

what is 12.5 times 0.74 and can you please put an explanation on how to get it please

Answers

9.25 the answer .... .... .... .... .... .... .... .... .... .... .... ....

Which of the expressions are equivalent to the one below? Check all that
apply.
(1 + 3) = 4
A. (3+1) = 4
B. 1. (3 + 4)
C. (3 + 4). (1+4)
D. (1 + 4) = 3

Answers

Answer:

A

Step-by-step explanation:

The answer to the first expression is 1.

A =1

B= 7

C=28

D=5/3

A and I think D, but I’m not exactly sure. I know for a fact that A matches up, but I think D is like 1.66666 something.

I need help with this ​

Answers

Answer:

-3=8

Step-by-step explanation:

y intercept is 6 explanation:

Given the sequence: -1, 2, 5, 8 what is a(2)?

Answers

Answer:

Your just going to keep adding 3 and the 2 is there because you had negative -1 and then you added 3 so hope this helps probably not what you looking for but hope this helps. Have a good day.

Step-by-step explanation:

Write a compound inequality for the given number line.

Answers

I think it is 1 > x ≥ 3. Sorry if I’m wrong, have a great day!

Can y’all help me here ?

Answers

Answer:

109

Step-by-step explanation:

3^2=9   -10^2=100

4x-2=-6
how to solve this math problem​

Answers

x=-1 is the answer to your question

Answer:8

Step-by-step explanation:  First you times 4 x 2 like

4+4=8 here is how i got the four you have to put the four 2 times i hope this helped

What is a decimal that is equivalent to the expression (3 x 10) + (4•1) + ( 7 • 1/10) + ( 5• 1/100)

Answers

The decimal form of the answer to that expression is

34.75

Help me out on this assignment?​

Answers

Answer:

#2 & #4

Step-by-step explanation:

So first lets plug this into point slope form which is

y-y₁=m(x-x₁)

So given the points (1,-7) and the slope -1, we can plug it into this equation

y+7=-1(x-1)

y+7=1x+1

y=1x-6

So you're answers to click would be, #2 & #4

One number is eighteen less than the second number, their sum is forty-eight, find both numbers.

Answers

Answer:

15 and 33

Step-by-step explanation:

brainliest would be great

What is the product of 50x0.08

Answers

The awnser is 4!Hope this helped you

Which equation could be used to find the value of x in the triangle below? (pls help)

Answers

Answer:

B.

Step-by-step explanation:

All of it needs to add up to 180

So you have

2x + 4x+10 + 6x-10 = 180

Combine like terms

12x=180

B- 12x = 180 because all of the numbers need to equal to 180

please I really need help
correct answer=branliest
1-more than/less than/the same as
2-greatest x -value/x-intercept/y-intercept/slope
3- greater than/less than/equal to
4-greatest x -value/x-intercept/y-intercept/slope
5-dollars per ounce/ounces per dollar​

Answers

Answer:

Gold is LESS expensive than platinym because SLOPE of graph is LESS

SLOPE gives UNIT DOLLARS PER OUNCE.

Step-by-step explanation:

sorry im late

Let f(x)=x2-3x + 9. Find f(-1).

Answers

X^2 - 3x + 9
(-1)^2 - 3(-1) + 9
1 - 3(-1) + 9
1 + 3 + 9
13

ahh help me please! I’m doing a test so right answers only please! tysm.

Answers

Answer:

The answer is B.

Step-by-step explanation:

Matthew went shopping for a new video game because of a sale. The price on the tag was $24, but Matthew paid $18 before tax. Find the percent discount.

Answers

Answer:

25%

Step-by-step explanation:

HELP ANSWER GETS BRAINLIEST
What is the simplified version for 2^2×2^3/2^4?

A 2^0

B 2^1

C 2^2

D 2^3

​

Answers

sawarasenaikimiwa⛓shoujonano?✨böKùWâÿARiçHiñBįCChīńOoSûDàYO

The answer is c I think I am not really sure
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